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Two-way transducers are closed under pre-composition with rational functions

Lax916827.TwoWayRationalPrecomposition · concepts/Lax916827/TwoWayRationalPrecomposition.lean · lax-916827

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    Natural Language Statement

    Theorem

    Functions computed by two-way transducers are closed under pre-composition with rational functions (Corollary C.2.7 of Transducers). By Theorem B.2.6 a rational function is a composition of prime Mealy machines, their right-to-left variants, homomorphisms and the separator function; Lemma C.2.6 handles the Mealy machines, a right-to-left machine is handled symmetrically, and a homomorphism is handled by simulating the head inside the image of a letter.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    1import Lax132576.RationalFunctions
    2import Lax916827.TwoWayTransducers
    3
    4/-!
    5---
    6title: Two-way transducers are closed under pre-composition with rational functions
    7type: theorem
    8---
    9Functions computed by two-way transducers are closed under pre-composition with
    10rational functions (Corollary C.2.7 of *Transducers*). By Theorem B.2.6 a
    11rational function is a composition of prime Mealy machines, their right-to-left
    12variants, homomorphisms and the separator function; Lemma C.2.6 handles the
    13Mealy machines, a right-to-left machine is handled symmetrically, and a
    14homomorphism is handled by simulating the head inside the image of a letter.
    15
    16# Formalization notes
    17
    18All three alphabets are assumed finite, as in the book.
    19-/
    20
    21namespace Lax916827.TwoWayRationalPrecomposition
    22
    23open Lax132576.RationalFunctions Lax916827.TwoWayTransducers
    24
    25/-- Pre-composing a two-way transducer with a rational function gives a two-way
    26transducer. -/
    27axiom isTwoWay_comp_isRationalFun {A B C : Type} [Finite A] [Finite B] [Finite C]
    28 {f : List A → List B} {g : List B → List C} (hf : IsRationalFun f) (hg : IsTwoWay g) :
    29 IsTwoWay (g ∘ f)
    30
    31end Lax916827.TwoWayRationalPrecomposition
    32
    Show Proof
    Formalization notes

    All three alphabets are assumed finite, as in the book.

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